Alice and Bob are sharing a cake. First Alice divides the cake into two pieces, then Bob chooses one piece, leaving Alice with the other piece. Alice can either divide the cake in half (a ratio of 1 : 1) or in a 1 : 2 ratio. Modeling this as a game gives the following game tree. For each of the following scenarios, write down the value for each of the three non-terminal nodes of the game treeIn other words, write down the utility that Alice should expect to receive in each of the game states given knowledge of the player strategies described below. Suppose both Alice and Bob are playing adversarially, each trying to maximize the amount of cake they receive. Suppose Alice still tries to maximize her share of the cake, but now Bob plays collaboratively, working to also help Alice get as much of the cake as possible. Suppose Alice still tries to maximize her share of the cake, but now Bob plays randomly, choosing the larger piece of cake with probability p. The state values will now be expressions involving p. How should Alice divide the cake (i.e. what action should she take at the root node) according to the value of p?
Alice and Bob are sharing a cake. First Alice divides the cake into two pieces, then Bob chooses one piece, leaving Alice with the other piece. Alice can either divide the cake in half (a ratio of 1 : 1) or in a 1 : 2 ratio. Modeling this as a game gives the following game tree.
For each of the following scenarios, write down the value for each of the three non-terminal nodes of the game treeIn other words, write down the utility that Alice should expect to receive in each of the game states given knowledge of the player strategies described below.
Suppose both Alice and Bob are playing adversarially, each trying to maximize the amount of cake they receive.
Suppose Alice still tries to maximize her share of the cake, but now Bob plays collaboratively, working to also help Alice get as much of the cake as possible.
Suppose Alice still tries to maximize her share of the cake, but now Bob plays randomly, choosing the larger piece of cake with probability p. The state values will now be expressions involving p. How should Alice divide the cake (i.e. what action should she take at the root node) according to the value of p?
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